So sorry to ask this, so far beneath most of your expertise . . .
Here goes: how tight should the plastic wrap be on the starter? Tight? Loose? Rubber-banded? Maybe a lid?
Or you could use dampened cheesecloth secured with an elastic band.
I use a regular pint sized round mason jar (no corners and no lip!) and fill it with the just-fed starter wich is about 1/4c, plenty of room to expand plenty. When the starter is rising on the counter, I put the lid on it, mostly screwed on so gas isn't totally trapped but the lid isn't going to come off by accident and no flying kitchen debris will fall in either. When I put the starter in the fridge, I tighten the lid.
I use a plastic tub with some small holes poked in the lid
I use regular glass cup for the starter, and insert the whole cup into re-sealable sandwich plastic bags, and seal it. it is a wonderful proofing environment.
I use a glass jar like this, but without rubber (or silicone, if it's a new verson) seal and without closing it tightly with the metal clip, just letting the lid sit losely on top.
There needs to be some air circulation for outgassing. Whatever lid you put on-paper,metal,glass,cloth-just make sure it is not too airtight.
The beauty of canning jars is that they behave as a one way valve. That is; expansion from the inside causes air to pass the rubber seal and escape. Once that expansion is finished, air is unable to re-enter the jar pass that closed rubber seal. This is why canning jars work as canning jars.
I keep my starter in a tightly closed canning jar with a rubber lined lid in place. Some pressure will remain in the jar and you will hear the escaping air when you open the jar.
I keep my sourdough in a closed Tupperware container in the refrigerator, to keep it from drying out or picking up bad odors. When it is out on the counter-top, I remove the lid and cover the container with a cotton handkerchief secured with a rubberband.
I used to keep all of my fermentation cultures on the counter-top in jars with loosely screwed on lids, but found that the kefirs in particular were changing their flavors. Now I keep all of them covered with just handkerchiefs while they are at room temperature. I am not sure why it was happening, except that perhaps foreign bacteria were coming in with the fresh air when it was unfiltered by cloth. I can definitely say that covering with only a fine plastic mesh will change a kefir culture. I left one covered that way in a room where I was manually milling some flour and it picked up wheat culture from the very fine flour dust in the room.